The Definite Integral

Key Questions

  • Answer:

    I don't have a one sentence answer

    Explanation:

    Above the x-axis
    If #f(x)# is non-negative over the interval #[a,b]#, then
    #int_a^b f(x) dx# is the area of the region between the graph and the #x#-axis between #x=a# and #x=b#

    Below the x-axis
    If #f(x)# is non-positive over the interval #[a,b]#, then
    #int_a^b f(x) dx# is the negative of area of the region between the graph and the #x#-axis between #x=a# and #x=b#

    Example:

    #int_0^3 sqrt(9-x^2) dx#.

    #a=0# and #b=3#.
    #f(x) = sqrt(9-x^2)# is never negative, so it is not negative on #[0,3]#

    Therefore the integral is equal to the area under the curve and above the #x#-axis between #x=0# and #x=3#.

    The graph of #y = sqrt(9-x^2)# is the part of #y^2 = 9-x^2# that has non-negative #y#-values. It is the upper semicircle for #x^2+y^2 = 9#
    The part between #x=0# and #x=3# is a quarter of a circle with radius 3.

    graph{y = sqrt(9-x^2)*(sqrt(1.5^2-(x-1.5)^2))/(sqrt(1.5^2-(x-1.5)^2)) [-2.63, 6.137, -0.812, 3.572]}

    So
    #int_0^3 sqrt(9-x^2) dx# is #1/4# of the area of the circle with radius #3#
    #int_0^3 sqrt(9-x^2) dx = (9 pi)/4#

  • Answer:

    An indefinite integral of a function #f(x)# is a family of functions #g(x)# for which: #g'(x)=f(x)#

    Explanation:

    An indefinite integral of a function #f(x)# is a family of functions #g(x)# for which: #g'(x)=f(x)#.

    Examples:

    1) if #f(x)=x^3#, then indefinite integral is:

    #int x^3dx=x^4/4+C#, because:

    #(x^4/4)'=4*x^3/4=x^3#, and #C'=0# for any real constant #C#

    2) If #f(x)=cosx#, then

    #int cosx dx= sinx+C#, because:

    #(sinx)'=cosx#

  • A definite integral is when you integrate a function over a specified interval. When completed you have a definite answer.

    Definite Integral because it is bounded

    #int_0^1 3xdx# evaluates to #[(3(1)^2)/2 - (3(0)^2)/2] = 3/2 - 0 = 3/2=1.5#

    Indefinite Integral because it is not bounded

    #int 3x dx# evaluates to #(3x^2)/2 + C#

Questions